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Full text of "Treatise On Analysis Vol-Ii"

3    THE MODULUS FUNCTION ON A GROUP        263

(c)    Suppose that Y is a normal subgroup of G. Then the measure (JL is (up to a
constant factor) the image of jnx  JU-Y under the homeomorphism (x, y) i > xy of
X x Y onto O, and for each x e X and jy e Y we have AG(x>>) = Ax(;c) AY(.K) mod(/x),
where ix is the automorphism v\~- *x~1vx of Y (cf. (14.4.6)).

(d)    Consider the locally compact space G = R x R*, with the law of composition
(x, y)(x', y') = (yxf -f- *, X/)- Show that this locally compact group is not unimodular
(use (c)).

(e)    Let G be a locally compact group. On the locally compact space E = R x G, a
law of composition is defined  by the formula (,*)(', xf) = (f+ AG(je)', xx').
Show that, with respect to this law of composition, E is a unimodular locally compact
group (use (b)). The group G (which is not necessarily unimodular) is isomorphic to
a subgroup and to a quotient group of E.

6.    Letp be a prime number. On the group Zp (Section 12.9, Problem 4) we define a ring
structure as follows: if z(zn) and z' = (z), then zz'=(znz'n) (the Z/p"Z being
quotient rings of Z). The canonical injection of Z into Zp (he. cit.) is a ring homo-
morphism, which identifies Z with a dense subring of Zp . Show that Zp is an integral
domain. Its field of fractions is called the field ofp~adic numbers and is denoted by
Qp . The fundamental system of neighborhoods of 0 in Zp formed by the balls with
center 0 is also a fundamental system of neighborhoods of 0 in Qp for a topology
compatible with the additive group structure of Qp . With respect to this topology,
Qp is separable, metrizable and locally compact (Section 12.8, Problem 1). The field
Q of rational numbers is dense in Qp, and the p-adic distance on Q (3.2.6) extends
to a distance d on Qp defining the topology of Qp . For z e Qp we write |z|p = d(0, z).
For each seQp, show that the modulus of the homothety z\~*sz of Qp is \s\p.
Deduce that, for each automorphism u of the vector space Qp over Qp, we have
mod(w) = |det |p.

7.    Let An denote Lebesgue measure in the space R" endowed with the Euclidean scalar

product (x | y) =  ji]j  For every universally measurable bounded set A, let crp(A)

be the set denned as follows. Identify R" with the product Rp x Rn~p, and for each
point x'epri(A) let Bn-p(x') be the closed Euclidean ball in RM~P with center 0
such that the measure An_p(Bn_p(;O) is equal to the measure Xn-p(A(x')) of the section
of A at x'. Then ap(A) is defined to be the union of the sets {x'} x Bn_p(x') as xf
runs through pri(A). Hence pr1(<rp(A)) = pri(A).

(a)    Show that if A is compact, then so is crp(A). (Use Problem 16(b) of Section 13.21.)
Deduce that if A is universally measurable, then crp(A) is An-measurable and that
An(<jp(A)) = Afl(A).

(b)    Show that if A and B are universally measurable bounded sets in R", then

An(ap(A) n Op(B)) = An(A n B) + An(o-p(A n CB) n ap(B n CA)).

(c)    Show that the mapping A i > crp(A) is not continuous with respect to the topology
on the set of nonempty bounded closed subsets of R" defined by the distance of Section
3.16, Problem 3. (Observe that in this topology every compact set can be approximated
arbitrarily closely by a finite set, which is of measure zero.)

(d)    If A, B are compact subsets of R", show that

aB.1(A) 4- ow^B) ^ an_1(A 4- B)/ < 0.